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L10a26

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Acknowledgement

L10a26 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X20,13,5,14 X14,7,15,8 X8,19,9,20 X16,10,17,9 X18,16,19,15 X10,18,11,17 X2536 X4,11,1,12

Gauss Code: {{1, -9, 2, -10}, {9, -1, 4, -5, 6, -8, 10, -2, 3, -4, 7, -6, 8, -7, 5, -3}}

Jones Polynomial: q-15/2 - 3q-13/2 + 6q-11/2 - 10q-9/2 + 12q-7/2 - 15q-5/2 + 13q-3/2 - 11q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant: - q-24 - q-22 + q-20 - 2q-18 + 2q-16 + 4q-14 + 5q-10 + q-8 + 2q-6 + q-4 - 3q-2 + 2 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial: a-1z3 + az-1 + 2az - az5 - 4a3z-1 - 9a3z - 6a3z3 - 2a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial: - a-2z4 + 2a-1z3 - 4a-1z5 + 1 - 3z2 + 9z4 - 8z6 - az-1 + 3az - 7az3 + 12az5 - 9az7 + 4a2 - 14a2z2 + 20a2z4 - 4a2z6 - 5a2z8 - 4a3z-1 + 15a3z - 34a3z3 + 41a3z5 - 16a3z7 - a3z9 + 7a4 - 21a4z2 + 16a4z4 + 8a4z6 - 8a4z8 - 4a5z-1 + 16a5z - 34a5z3 + 34a5z5 - 10a5z7 - a5z9 + 4a6 - 13a6z2 + 9a6z4 + 3a6z6 - 3a6z8 - a7z-1 + 4a7z - 9a7z3 + 9a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 6          1
j = 4         3 
j = 2        51 
j = 0       63  
j = -2      86   
j = -4     75    
j = -6    58     
j = -8   57      
j = -10  26       
j = -12 14        
j = -14 2         
j = -161          


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
Length[Skeleton[L]]
Out[2]=   
2
In[3]:=
Show[DrawMorseLink[Link[10, Alternating, 26]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 26]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 5, 14], X[14, 7, 15, 8], 
 
>   X[8, 19, 9, 20], X[16, 10, 17, 9], X[18, 16, 19, 15], X[10, 18, 11, 17], 
 
>   X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 2, -10}, {9, -1, 4, -5, 6, -8, 10, -2, 3, -4, 7, -6, 8, -7, 
 
>    5, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     3       6      10     12     15     13      11
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
       3/2    5/2
>   4 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -22    -20    2     2     4     5     -8   2     -4   3       2
2 - q    - q    + q    - --- + --- + --- + --- + q   + -- + q   - -- - 3 q  + 
                          18    16    14    10          6          2
                         q     q     q     q           q          q
 
       6    8
>   2 q  - q
In[8]:=
HOMFLYPT[Link[10, Alternating, 26]][a, z]
Out[8]=   
       3      5    7                                     3
a   4 a    4 a    a               3        5      7     z       3  3
- - ---- + ---- - -- + 2 a z - 9 a  z + 6 a  z - a  z + -- - 6 a  z  + 
z    z      z     z                                     a
 
       5  3      5      3  5
>   3 a  z  - a z  - 2 a  z
In[9]:=
Kauffman[Link[10, Alternating, 26]][a, z]
Out[9]=   
                                     3      5    7
       2      4      6    8   a   4 a    4 a    a                3
1 + 4 a  + 7 a  + 4 a  + a  - - - ---- - ---- - -- + 3 a z + 15 a  z + 
                              z    z      z     z
 
                                                                            3
        5        7        2       2  2       4  2       6  2      8  2   2 z
>   16 a  z + 4 a  z - 3 z  - 14 a  z  - 21 a  z  - 13 a  z  - 3 a  z  + ---- - 
                                                                          a
 
                                                     4
         3       3  3       5  3      7  3      4   z        2  4       4  4
>   7 a z  - 34 a  z  - 34 a  z  - 9 a  z  + 9 z  - -- + 20 a  z  + 16 a  z  + 
                                                     2
                                                    a
 
                           5
       6  4      8  4   4 z          5       3  5       5  5      7  5      6
>   9 a  z  + 3 a  z  - ---- + 12 a z  + 41 a  z  + 34 a  z  + 9 a  z  - 8 z  - 
                         a
 
       2  6      4  6      6  6    8  6        7       3  7       5  7
>   4 a  z  + 8 a  z  + 3 a  z  - a  z  - 9 a z  - 16 a  z  - 10 a  z  - 
 
       7  7      2  8      4  8      6  8    3  9    5  9
>   3 a  z  - 5 a  z  - 8 a  z  - 3 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    6      1        2        1        4        2        6        5       7
6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
     2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
    q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      5       8       7      5      8              2      2  2      4  2    6  3
>   ----- + ----- + ----- + ---- + ---- + 3 t + 5 q  t + q  t  + 3 q  t  + q  t
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t


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